$THH$ and base-change for Galois extensions of ring spectra

TL;DR

This paper investigates the behavior of topological Hochschild homology (THH) under base-change for Galois extensions of ring spectra, providing new positive results, examples, and a counterexample.

Contribution

It extends previous results on THH base-change for Galois extensions and introduces new examples and a counterexample demonstrating when base-change holds or fails.

Findings

Base-change holds for certain Galois extensions, extending prior results.

New examples of Galois extensions where base-change is valid.

A counterexample showing base-change can fail in some cases.

Abstract

We treat the question of base-change in $THH$ for faithful Galois extensions of ring spectra in the sense of Rognes. Given a faithful Galois extension $A \to B$ of ring spectra, we consider whether the map $THH(A) \otimes_A B \to THH(B)$ is an equivalence. We reprove and extend positive results of Weibel-Geller and McCarthy-Minasian and offer new examples of Galois extensions for which base-change holds. We also provide a counterexample where base-change fails.